Lecture 2: Erasures, Errors and Begining to Combat Them

You can probably envision several different ways that the note from your mom that we atarted with can fail to get the message to you. Let's assume that you did indeed get the message, just that it could come out wrong. There are two big types of things that could go wrong with this note.

You could get the note, but with some part of it illegible, perhaps a glass of water spilled on it and smudged one of the words so that you can't read it anymore, this is called an erasure. You just got nonsense instead of one of the words, so it's not like you think that you you got one message and your really got a different one it's merely that you don't know what the word was at all.

Alternately, maybe you have a pesky little brother who came along and changed some of the letters in your mom's note to other letters so surreptitously that you can't tell the handwriting on his new word from the one that your mom would have written. This is a different sort of problem in transmission of the message called an error.

So what might you do upon getting a message from your mom with an erasure that reads "Don't be late for dinner *smudge* Aunt Gertrude will be there." Does the message still get across? What do you think that the smudge should be? How did you figure out what the smudge should say so that you still could do as your mom asked?

On a differnt note, what happens when your brother changes some of the letter in the note randomly (he's fair enough not to change the note specifically to get you in trouble) and you get "Don't fe wate for dinner Xunt Gertrude will be qhere." Can you make sense of this note? What allows you to make sense of something that seems like nonsense?

In both of the cases above, you were probably able to make sense of the note from your mom, despite the intervention of a glass of water or a little brother because the note should make sense, it should be something that is English, and you can probably fill in what the something should be if it is a little off, just as you might correct a spelling error in a friend's paper. In some sense expecting that the message that you get should be comprehensible is perhaps the most basic error correcting code. It was applied often in the age of the telegraph. Telegrams has errors in them a lot, but since the people recieving them knew that they had to make sense, they could usually be interpreted to mean what they were meant to say, just as you could interpret the notes from your mother.

Let's talk now about the task of transmitting 0s and 1s that might be subject o erasures and errors. Let's start by trying to transmit a string of only one 1. So our message is just 1. Our old idea of using the fact that the message should be sensible Engliish doesn't apply here, since 1 doesn't have much of any meaning to us right now, so we need a new method of protecting against error and erasures in the transmission. One choice might be just to send the message again, so that there is a larger chance of the message being recieved properly, or at least not recieved as the opposite of what the sender wants. In fact, if we send the message 1 a bunch of times times we can take a majority vote in the end of the messages recieved and decide what message the original sender was trying to send.

Let assume that our message of 1 can only become erased, not get any errors. How many erasures can the code that repeats the message 3 times have and still get the message across? What about a code that repeats the message 4 times? What about 763 times? n times?

Now let's assume that your meddling little brother is back on duty changing things, but that no erasures are possible. We are going to decide what the original message was by a majority vote of the messages recieved. If we send 3 copies of the message 1, we can get any strong of three 0s and 1s with the right set of erros. Those possible strings are: 000, 001, 010, 100, 011, 101, 110, 111. Which of these actually transmit the right message when we take the majority vote to determine what the message was? Can you do the same analysis for sending 4 copies of the message? By thinking about what it means to win a majority vote, can you say how many errors are allowed if we send n copies of the message and still want to get the correct message?

Hints and Solutions:

Does the message still get across? What do you think that the smudge should be? How did you figure out what the smudge should say so that you still could do as your mom asked? YOu would probably still get the message from your mom, after all, you are probably pretty sure that your mom is still communicating in English, and that she wrote a reasonable English sentance, so you could probably figure out to replace the smudge with because.

Can you make sense of this note? What allows you to make sense of something that seems like nonsense? Again, you can probably make sense of this note by realizing again that your mom is trying to speak English and, since you know English too, turn the note into something completely sensible.

How many erasures can the code that repeats the message 3 times have and still get the message across? What about a code that repeats the message 4 times? What about 763 times? n times? We just always need one 1 remaining, the rest can be erased.

Which of these actually transmit the right message when we take the majority vote to determine what the message was? Can you do the same analysis for sending 4 copies of the message? By thinking about what it means to win a majority vote, can you say how many errors are allowed if we send n copies of the message and still want to get the correct message? We need a more than half of the recieved items in the string to be 1.